# Tagged Questions

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### Stochastic Integrals and Martingales

I am attempting the following proof but two aspects of the solution confuse me: Given \begin{align} I^{n}_{t} = \int^t_0 \Delta_u^ndW_u = \sum_{j=0}^{k-1}\Delta_{t_{j}}(W_{t_{j+1}}-W_{t_{j}}) + ...
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### Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
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### Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
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### Expectation of a stochastic integral conditioned on a particular σ-algebra

Suppose that $g$ is a simple process in the class $\mathcal{V}=\mathcal{V}[U,T]$. Using the notations $g_k=g(t_k)$, $\Delta B_k = B(t_{k+1})-B(t_k)$, and $\mathcal{F}_k=\mathcal{F}_{t_k}$, with the ...
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### Expectation of a stochastic integral

Let $M$ be a right-continuous local martingale, $s,t$ two times (stopping times, if you like). Under what conditions does the following hold: $$E\left(\int_s^t X \, dM\mid\mathcal{F}_s\right)\le 0$$ ...
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### Defining the Radon-Nikodym as a solution to an SDE

Can someone please clarify this to me: If I have the Radon-Nikodym $L_t=\frac{dQ}{dP}$, on $\mathcal{F}_t$, then I know that $L_t$ is a non-negative P-martingale. So in many textbooks they say it is ...
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### Canonical semimartigale truncation function meaning

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: $H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$M_t = \sum_{0<s\leq t} ... 1answer 158 views ### Preservation of Martingale property Can someone help me to prove this? If possible I'd like the prove can avoid the use of local martingale. Prove the Ito integral \int_0^T \Delta_t(\omega) dW_t(\omega) is a martingale if E[\int_0^T ... 1answer 137 views ### Girsanov transformation and preservation of independence If we create a weak solution of an SDE using the Girsanov transformation, are the initial condition and parameters independent of the transformed Wiener process if they are independent of the original ... 0answers 46 views ### Supermartingale Lemma + related problems Given the following Lemma: Let A_{t}=\int_{0}^{t}a_{s}dB_{s} where a is an adapted process satisfying \mathbb{P}\Big(\int_{0}^{T}a^{2}_{u}du < \infty\Big) = 1 and B is a standard Brownian ... 1answer 106 views ### Martingale inequality Let f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R} be a deterministic function, as nice as you want, W a Brownian motion and define$$ Y^r_t := \int_0^t f(r,s) dW_s $$For each fixed r, ... 1answer 316 views ### Show that this continuous local martingale is a martingale We are given the following SDE:$$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$and$$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$We are asked to apply Ito's formula to F(t,X_t) for t\geq0 ... 0answers 76 views ### Local martingale iff each component is a local martingale? This is probably an easy question: A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to \infty) such that the stopped process is a ... 1answer 395 views ### \mathcal{F_t}-martingales with Itô's formula? I need a little help with a problem. I am given some stochastic processes and supposed to show that they are \mathcal{F_t}-martingales. The first one is this, and they all look similar: ... 2answers 180 views ### Is the solution to a driftless SDE with Lipschitz variation a martingale? If \sigma is Lipschitz, with Lipschitz constant K, and (X_t)_{t\geq 0} solves$$dX_t=\sigma(X_t)dB_t,$$where B is a Brownian motion, then is X a martingale? I'm having difficulty getting ... 2answers 113 views ### If X is a martingale, X(0)=0; f left continuous, is \int f X dt also a martingale? If X(t) is a martingale, and X(0) = 0. f(t) is a left continuous function,$$ g(t) = \int_0^t f(s) X(s) ds $$is g(t) also a martingale? I guess it shall be, but don't know how to prove ... 1answer 326 views ### d-Dimensional Brownian Motion Martingales Let d > 1 and let W_t denote a standard d-dimensional Brownian motion starting at x\neq 0. Let M_t = \log|W_t| for d = 2, and M_t= |W_t|^{2-d} for d > 2. Show that M_t is a ... 1answer 519 views ### Martingale problem If X_t is an \mathbb{R}- valued stochastic process with continuous paths, show that the following two conditions are equivalent: (i) for all f\in C^2(\mathbb{R}) the process$$f(X_t) − f(X_0) ...
Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution  ...